Graphics Reference
In-Depth Information
Fig. 2.4
The product of two complex numbers
cos
(θ
+
φ)
=
cos
θ
cos
φ
−
sin
θ
sin
φ
+
=
+
sin
(θ
φ)
sin
θ
cos
φ
cos
θ
sin
φ
φ)
.
So the product of two complex numbers creates a third one with modulus
|
rs
cos
(θ
=
+
+
+
zw
φ)
i
sin
(θ
zw
|=
rs
and argument
arg
(zw)
=
arg
(z)
+
arg
(w)
=
θ
+
φ.
Let's illustrate this with an example. Figure
2.4
shows two complex numbers
z
=
1
+
i,
w
=
2
i
therefore,
√
2
,
|
|=
=
z
arg
(z)
45°
|
w
|=
2
,
arg
(w)
=
90°
2
√
2
|
zw
|=
arg
(zw)
=
135°
which is another complex number
−
2
+
2
i
.
2.12 Rotors
The above observations imply that multiplying a complex number by another, whose
modulus is unity, causes no scaling. For example, multiplying 3
0
i
creates the same complex number, unscaled and unrotated. However, multiplying
3
+
4
i
by 1
+
+
4
i
by 0
+
i
rotates it by 90° without any scaling.
